A note on a new class of generalized Pearson distribution arising from Michaelis-Menten function of enzyme kinetics

Many problems of enzyme kinetics can be described by a function known as the Michaelis-Menten (M-M) function. In this paper, motivated by the importance of Michaelis-Menten function in biochemistry and other biological phenomena, we have introduced a new class of generalized Pearson distribution arising from Michaelis-Menten function. Various properties of this distribution are derived, for example, its probability density function (pdf), cumulative distribution function (cdf), moment, entropy function, and relationships with some well-known continuous probability distributions. The graphs of the pdf and cdf of our new distribution are provided for some selected values of the parameters. It is observed that our new distribution is positively skewed and unimodal. We hope that the findings of this paper will be useful in many applied research problems. 2000 Mathematics Subject Classification: 60E05, 62E10, 62E15.


Introduction
The Michaelis-Menten function is one of the most important mathematical functions to model many problems of biochemistry and other biological phenomena to describe enzymatic reactions, see, for example, Michaelis and Menten [13], Briggs and Haldane [3], Cleland [5], Fontes et.al. [8], [9], among others. It is defined by the following equation: [] where v is the initial velocity in an enzyme-catalyzed reaction, max V is the maximal velocity, i.e. the velocity attended at very high concentration of substrate [S], m K is the Michaelis constant and corresponds to the concentration of substrate at which max /2 vV  . As pointed out by, Fontes et.al. [9], the Michaelis-Menten equation (1) can be reduced to the following forms known as Type I, Type II and Type III respectively: where the constants have their usual meanings. For detailed mathematical analysis and applications to enzymatic reactions of the above three types of the Michaelis-Menten equation, see Fontes et.al. [9], among others. In this paper, motivated by the importance of Michaelis-Menten function in biochemistry and other biological phenomena, we have introduced a new class of generalized Pearson distribution arising from Michaelis-Menten function. The organization of this paper is as follows. Section 2 contains a review of existing classes of generalized Pearson continuous probability distributions as considered by various researchers. We have identified as many as 14 such distributions. Section 3 contains the derivations of the probability density function ( pdf ) and cumulative distribution function (cdf) of our proposed new class of generalized Pearson distribution arising from Michaelis-Menten function. In Section 4, some distributional properties of our proposed new class of generalized Pearson distribution, along with the graphs of the pdf and cdf for some selected values of the parameters, are provided. Section 5 contains some concluding remarks.

Review on existing classes of generalized Pearson system of distributions
A continuous probability distribution belongs to the Pearson system if, for a positive continuous random variable X , its probability density function ( pdf ) f satisfies a differential equation of the form where a , b , c , and d are real parameters such that f is a . pdf The shapes of the pdf depend on the values of these parameters, based on which Pearson [16,17] classified these distributions into a number of types known as Pearson Types I -VI. Later, in another paper, Pearson [18] defined more special cases and subtypes known as Pearson Types VII -XII. Many well-known distributions belong to these types of Pearson distributions which include Normal and Student's t distributions (Pearson Type VII), Beta distribution (Pearson Type I), Gamma distribution (Pearson Type III), among others. For details on the Pearson systems of continuous probability distributions, the interested readers are referred to Johnson et. al. [12]. In recent years, many researchers have considered a generalization of the Pearson system, known as generalized Pearson system of differential equation (GPE), given by where m , N n    /0 and the coefficients j a and j b are real parameters. The system of continuous univariate pdf s  generated by GPE is called a generalized Pearson system which includes a vast majority of continuous pdf s  by proper choices of these parameters. We have identified as many as 14 such distributions, which are provided below: i) Roy [21] studied GPE, when 0 2, 3, 0 m n b    , to derive five frequency curves whose parameters depend on the first seven population moments. ii) Dunning and Hanson [7] used GPE in his paper on generalized Pearson distributions and nonlinear programming. iii) Cobb et. al. [6] extended Pearson's class of distributions to generate multimodal distributions by taking the polynomial in the numerator of GPE of degree higher than one and the denominator, say,   vx, having one of the following forms: a) iv) Chaudhry and Ahmad [4] studied another class of generalized Pearson distributions when Lefevre et al. [13] studied characterization problems based on some generalized Pearson distributions.
vi) Considering the following class of GPE   2 0 Sankaran [22] proposed a new class of probability distributions and established some characterization results based on a relationship between the failure rate and the conditional moments. vii) Stavroyiannis and Stavroulakis [28] studied generalized Pearson distributions in the context of the superstatistics with non-linear forces and various distributions.
viii) Rossani and Scarfone [20] have studied GPE in the following form   , and used it to generate generalized Pearson distributions in order to study charged particles interacting with an electric and/or a magnetic field. ix) Shakil et. al. [24] defined a new class of generalized Pearson distributions based on the following differential which is a special case of the GPE (2), when 2, 1 mn  , and 0 0 b  . The solution to the differential equation (3) is where where () p Dzdenotes the well-known parabolic cylinder function.
x) Shakil and Kibria [23] consider the GPE (2) in the following form The solution to the differential equation (6) is given by where The solution to the differential equation (9) is given by where   2 p K   denotes the well-known modified Bessel function of third kind. For the characterizations of the above continuous probability distribution, due to Shakil et. al. [25], known in the literature as the Shakil-Kibria-Singh (SKS) distribution, the interested readers are referred to Hamedani [11] where the the Shakil-Kibria-Singh (SKS) distribution has been characterized by Hamedani [11] based on a simple relationship between two truncated moments, and the hazard function. xii) Hamedani [11] has defined a new variation of the continuous probability distribution (10) in a bounded domain. The pdf of Hamedani's distribution is given by where 0   , 0   , and 0 p  are parameters and is the normalizing constant.
The cdf corresponding to the pdf (12) is given by For the special case of   , we have where 0   and 0 p  are parameters. As pointed out by Hamedani [11], the pdf f given by (12) satisfies the which is a special case of GPE (2). For the characterization of the pdf in Eq. (12), when N p    /0, the interested readers are referred to Hamedani [11]. xiii) Ahsanullah et. al. [2] defined a new class of distributions as solutions of the GPE (2). They considered the following differential equation which is a special case of the generalized Pearson Eq. (2) when 2, where we assume that β > 0, γ > 0, 0 < ν < 1, 0 < µ < 1, 1 -µ > ν > 0. Integrating the above equation, we have Using the equation ( where , () ab Wz denotes the well-known Whittaker function (see Abramowitz and Stegun [1], page 505, chapter 13).
xiv) Recently, Stavroyiannis [27] defined a new class of distributions as solutions of the GPE (2) by considering the following differential equation which is a special case of the generalized Pearson Eq. (2) when 5, 6 mn  . By taking special values of the coefficients j a and j b , Stavroyiannis [27] obtained the GPE in the following form  with its solution given by the following probability density function: where  is the location parameter, 0 a  is the scale parameter, 1 2 m  and 0 b  control the kurtosis,  is the asymmetry parameter, and C is the normalization constant. As pointed by Stavroyiannis [27], the above distribution with the pdf (19) includes an extra fourth order term in the denominator to account for fat and thick-tails for the case of 0 b  . The distribution becomes double peaked for the case of a negative b coefficient, while for 0 b  the Pearson-IV distribution is regained. For details on these, the interested readers are referred to Stavroyiannis [27].

A new class of generalized pearson distribution arising from michaelismenten function
In this section, for a positive continuous random variable X, we define a new class of generalized Pearson distributions based on the following differential equation where      , and C denotes the normalizing constant.

Expressions for the normalizing constant
In order that the right side of the Eq. (21) represents a probability density function (pdf), we must have  [19], the expression for the normalizing constant C is easily obtained, after simplification, as follows

Expression for the cumulative distribution function
where  

Distributional properties
In what follows, some properties of our proposed distribution are given below.

Graphs of the PDF and CDF
The  Figures 1-6. The effects of parameters can be easily seen from these graphs. Also, it is clear from these graphs that our proposed distributions of the random variable X are positively (that is, right) skewed and unimodal.

nth Moment
In what follows, we derive the moments of our proposed distribution. We have In Eq.  [19], the following expression for the nth moment is easily obtained: where C denotes the normalizing constant given by (24),  , of our proposed distribution which can be obtained by using the formula:

Shannon entropy
An entropy provides an excellent tool to quantify the amount of information (or uncertainty) contained in a random observation regarding its parent distribution (population). A large value of entropy implies the greater uncertainty in the data. As proposed by Shannon [26], if X is a none-negative continuous random variable with pdf   X fx , then Shannon's entropy of X , denoted by   hf or   hX , is defined as Now, in Eq. (28) Above, using the pdf   fx of the our proposed distribution, as given in Eq. (21), and then integrating and simplifying, Shannon entropy of our proposed distribution is easily obtained as follows where C denotes the normalizing constant given by Eq. (24), and  

Distributional relationships
It is easy to see that, by a simple transformation of the variable x or by taking special values of the parameters      ) can be expressed as the pdf of the product of the pdf's of the exponential and some members of the family of Burr distributions (such as Lomax, or Pareto Type I, or Pareto Type II distributions).

Concluding remarks
In this paper, we have introduced a new class of generalized Pearson distribution arising from Michaelis-Menten function. Also, we have reviewed existing classes of continuous probability distributions which can be generated from the generalized Pearson system of differential equation (GPE), as given in Eq. (2). We have identified as many as fourteen such distributions. Various properties of our proposed distribution are derived, for example, its probability density function (pdf), cumulative distribution function (cdf), moment, entropy function, and relationships with some well-known continuous probability distributions. The graphs of the pdf and cdf of our proposed distribution are provided for some selected values of the parameters. It is observed that our proposed distribution is positively skewed and unimodal. We hope that the findings of this paper will be useful in many applied research problems. Some open problems and direction for future research for our proposed generalized Pearson distribution are characterization, estimation of parameters, applications to real world problems, Bayesian analysis, regression analysis, among others. Further, we hope that our proposed attempt will be helpful in designing a new approach of unifying different families of distributions based on the generalized Pearson